This is really a huge list of ideas..

My favourite so far is the AI-compo, I'd like to see some really intelligent enemies.
But I also have some ideas for an upcoming compo:
It's on the news today Compo: the main idea here is that every game looks like a news broadcast with an anchor and a reporter (remember the good old 'tv sports...'-series). The anchor introduces the player and the goal or the boss. The anchor talks to the reporter ("Peter, please report to us: what the hell is going on there?"), the camera zooms to the reporter screen, the reporter makes a statement and the game begins. It's possible to distract the player with split-screen commercials... ("We'll be right back after this commercial!")
Everyday life Compo: Pick an activity that you do often, e.g. brushing your teeth or mowing the lawn or going to school/work. What normally is boring should be interesting in a game...
Real bad and mean and ugly boss Compo: Make a game with a boss that is so bad and mean and ugly that you get nightmares... the boss really has to do mean things like ...ehh.. stealing candy from a baby. The game with the meanest boss wins this compo.
Movie trailer Compo: Make a game that looks like a movie trailer. The text of this trailer could be generated by the movie trailer generator here:
http://www.pecknology.net/trailergen/